A catchphrase is a short phrase that is widely recognized and repeated.
While somewhat difficult to define strictly, they are very much a part of our culture, and a few examples make the concept obvious, if no less difficult to define:
They are, essentially, linguistic earworms, a free pass to a conversational turn, and a way to share cultural connections. These can last for generations (or longer; e.g., let them eat cake), or can pass in a flash (on fleek), can be quite long (But, soft! what light through yonder window breaks?) or short (sad). Some just become part of the language; pass (as in to pass one's turn) was listed as a catchphrase coming to us from game shows in the 1995 Cambridge Encyclopedia of the English Language, but probably does not especially hold that context for most people now. Likewise, a large portion of our language started out as colorful and slangish linguistic tricks; unmentionable, critter, zillion, burgle, google, tongue-tied, and a million others could be argued to be catchphrases when they first appeared, although suddenly popular words are sometimes called vogue words rather than catchphrases.
Catchphrases overlap significantly with a number of other categories; you'll have noticed that many of these are lyrics, slogans, and slang. Most are very much tied to context, which includes place; you may not recognize that a phrase is a catchphrase if you aren't part of the ingroup, although the rise of broadcast media has made the spread of catchphrases ever more universal and uniform, with many spreading quickly across the English-speaking world; at this point in history language barriers largely contain catchphrases, although as translation services improve this may not be the case much longer.